Source-linked AI summary
Secrecy Outage Analysis over Correlated Composite Generalized-Gamma Fading Channels
Muhammad Adil, Pranavkumar Pathak, Salman A. Alqahtani
TL;DR
The paper addresses the limited flexibility of secrecy analyses that constrain one fading component to conventional Gamma behavior. It develops a correlated GG/GG framework using Mellin transforms and Fox-H functions, deriving closed-form and exact-series secrecy expressions. Numerical and Monte Carlo results validate the analysis and show significant effects of GG fading exponents and shadowing correlation on secrecy performance.
Problem
Existing secrecy analyses often restrict one propagation component to a conventional Gamma-based form, limiting modeling flexibility when both multipath fading and shadowing depart from Gamma behavior.
Method
The paper models independent GG small-scale fading with correlated GG shadowing and uses Mellin transforms and Fox-H functions to derive secrecy distributions and outage expressions.
Results
Numerical and Monte Carlo results validate the analysis and demonstrate significant effects of GG fading parameters and shadowing correlation on secrecy performance.
Takeaways & Limitations
The GG/GG framework provides a more flexible secrecy-outage characterization while encompassing Nakagami-m/GG and Nakagami-m/Gamma models as special cases.
Abstract
from arXiv · showhide
This paper investigates physical-layer security (PLS) over correlated composite generalized-Gamma (GG)/GG fading channels, where both shadowing and small-scale fading follow GG distributions. Using Mellin transforms and Fox-H functions, closed-form expressions are derived for the single-link probability density function (PDF), joint distribution, survival function, and zero-rate secrecy outage probability (SOP)/probability of non-zero secrecy capacity (PNZSC). The general-rate SOP is expressed as an exact double series with one residual onedimensional integral per term. The model includes the Nakagamim/GG and Nakagami-m/Gamma channels as special cases. Numerical results validate the analysis and demonstrate the impact of the fading parameters on secrecy performance.
I. INTRODUCTION
The paper addresses secrecy analysis when legitimate and eavesdropping links experience correlated composite fading whose shadowing and small-scale components both follow generalized-Gamma behavior. It develops a unified GG/GG framework, derives analytical secrecy metrics, and validates the model numerically.
- Motivation: Existing secrecy analyses often restrict one propagation component to a conventional Gamma-based model, limiting flexibility when both fading processes deviate from Gamma behavior.The limitation is relevant to dense urban, indoor, blockage-prone, and heterogeneous wireless environments.
- Channel model: The model uses independent GG small-scale fading on the legitimate and eavesdropping links with correlated shadowing between them.Only the shadowing pair is correlated; the small-scale components remain independent.
- Analytical framework: Mellin transforms and Fox-H functions yield closed-form single-link PDFs, joint distributions, survival functions, and zero-rate SOP/PNZSC expressions.These results form the analytical core of the proposed secrecy framework.
- Special cases: The framework contains Nakagami-m/GG and Nakagami-m/Gamma channels as special cases through successive unity settings of the GG exponents.This provides consistency with established composite-fading models.
- Analytical framework: The general-rate SOP is represented by an exact double series with one residual one-dimensional integral per term.The representation is retained because the efficient Nakagami-m/Gamma quadrature does not directly extend to the general GG/GG kernel.
- Validation: Numerical and Monte Carlo results validate the analysis and show that GG fading parameters and shadowing correlation affect secrecy outage performance.For equal average SNRs, outage increases with target secrecy rate and approaches unity as the rate rises.
A. Mellin Transform
The single-link analysis models received power as the product of independent generalized-Gamma shadowing and small-scale fading factors. Mellin-transform factorization exposes symmetric dependence on their respective shape and exponent parameters.
- Single-link construction: The single-link received-power variable is written as Z = BU, with B = Y^(1/β) and U = V^(1/κ).Y and V are independent Gamma random variables underlying the shadowing and small-scale components.
- Mellin-transform factorization: Independence of B and U allows the Mellin transform of Z to factor into the transforms of the shadowing and small-scale factors.This factorization is the key simplification used in the single-link derivation.
- Parameter structure: The resulting expression depends symmetrically on shadowing parameters (k, β) and small-scale fading parameters (m, κ).That symmetry is subsequently exploited in the Fox-H representation.
B. Fox-H Function Representation
The Fox-H representation converts the Mellin-transform result into a closed-form PDF for the single-link composite GG variable. The expression reduces to Nakagami-m/GG and Nakagami-m/Gamma forms under unity exponent settings.
- PDF representation: Proposition 1 gives the PDF of the single-link variable Z for z > 0 in Fox-H-function form.The representation follows by matching the Mellin transform with the Mellin–Barnes identity.
- Fox-H notation: The notation separates upper and lower parameter pairs in the standard Fox-H-function representation.The small-scale GG exponent is denoted κe to avoid clashing with the kernel symbol.
- Special cases: Setting κe = 1 reduces the PDF to the Nakagami-m/generalized-Gamma model, while κe = β = 1 further gives the classical Nakagami-m/Gamma model.These reductions connect the generalized expression to established composite-fading cases.
IV. JOINT PDF UNDER CORRELATED GG SHADOWING AND INDEPENDENT GG SMALL-SCALE FADING
The joint model represents correlated shadowing through a double mixture while retaining independent GG small-scale fading, with normalized weights under a parameter-ordering condition.
- Only the shadowing pair is correlated; the small-scale fading variables remain mutually independent and independent of shadowing.
- The correlated-Gamma expansion becomes a double mixture of normalized single-link kernels indexed by i and j.
- For kE ≥ kB > 0, the mixture weights are nonnegative and normalized.
- The resulting double-series representation is used under kE ≥ kB > 0; when kB > kE, the Gamma-pair indices must be interchanged.
V. SECRECY PERFORMANCE ANALYSIS
The secrecy-performance derivation obtains the composite survival function through Mellin–Barnes representations and contour integration, then identifies the result as a Fox H-function.
- The composite survival function is derived from the Mellin–Barnes representation of the underlying kernel.
- Choosing a contour with ℜ(ζ) > 1 and shifting variables produces the integral form used for the survival-function derivation.
- The Mellin–Barnes kernel matches a Fox H-function with specified upper and lower parameter pairs.
B. Closed-Form Zero-Rate Outage / PNZSC
The zero-rate secrecy analysis combines legitimate-link survival and eavesdropping-link PDF terms through Mellin transforms and Parseval integration, yielding a Fox-H closed form summed over mixture indices.
- B. Closed-Form Zero-Rate Outage / PNZSC: The convolution framework combines the legitimate-link survival function with the eavesdropping-link PDF in the Qij representation.
- B. Closed-Form Zero-Rate Outage / PNZSC: For each mixture term, latent indices define the effective shapes and link-specific kernels used in the convolution integral.
- B. Closed-Form Zero-Rate Outage / PNZSC: The Mellin transforms of the survival function and PDF are combined through Mellin–Parseval integration over a valid vertical contour.
- B. Closed-Form Zero-Rate Outage / PNZSC: The resulting parameter collection is identified with a Fox H-function and summed over normalized mixture weights.
- B. Closed-Form Zero-Rate Outage / PNZSC: The analysis evaluates secrecy outage performance for the proposed correlated composite GG/GG fading model.
C. Secrecy Outage Probability at a General Rate
At a general target rate, the secrecy outage expression uses a double-series expansion with one residual one-dimensional integral per term because the rate-dependent argument is affine rather than multiplicative.
- For fixed mixture indices, integrating the legitimate-link survival function over the eavesdropping-link PDF gives the conditioned secrecy-success probability.
- At r = 0, the double-series integral reduces to the PNZSC expression, so the outage probability becomes Po(0) = 1 − PNZSC.
- For r > 0, the Proposition 3 convolution cannot be applied directly because h(x2, r) is affine rather than purely multiplicative.
- The general-rate result is retained as an exact double-series representation with one residual one-dimensional integral per mixture term, evaluated numerically using Fox-H representations.
VI. NUMERICAL RESULTS AND DISCUSSION
The numerical results show that target-rate and zero-rate secrecy outage depend strongly on generalized shadowing, generalized small-scale fading, and shadowing correlation. The GG/GG model therefore provides a more flexible characterization than the Nakagami-m/Gamma baseline.
- The outage probability increases with target secrecy rate and approaches unity when legitimate and eavesdropping links have equal average SNR.The evaluated setting uses k1 = k2 = 1, m1 = m2 = 4, ρ = 0.6, and ¯γ1 = ¯γ2 = 4 dB.
- Both β and κ significantly affect secrecy performance across the baseline, GG-shadowing-only, GG-small-scale-only, and full GG/GG cases.
- Varying β produces clearly different zero-rate outage levels, confirming the importance of generalized shadowing.
- With β = 1 fixed, varying κ changes the zero-rate outage level and its decay with increasing ¯γ1.
- Varying ρ noticeably changes Po(0), highlighting correlated shadowing between the legitimate and eavesdropping links.
- Joint variation of β and κ enables the GG/GG model to describe a wider range of physical-layer security behaviors than the Nakagami-m/Gamma baseline.
VII. CONCLUSION
The paper develops a unified secrecy-outage framework for correlated composite GG/GG fading channels and derives analytical representations for key secrecy quantities. The model also contains established Nakagami-m-based channel models as special cases, with numerical and Monte Carlo results confirming the analysis and parameter effects.
- The framework derives closed-form expressions for the single-link PDF, joint distribution, survival function, and zero-rate SOP/PNZSC using Mellin transforms and Fox-H functions.
- The general-rate SOP is represented by an exact double series with one residual one-dimensional integral per term.
- The model includes Nakagami-m/generalized-Gamma and Nakagami-m/Gamma channels as special cases.
- Numerical and Monte Carlo results confirm the analysis and show significant effects of fading exponents and shadowing correlation on secrecy performance.